On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient

Impacto

Doubova, Anna and Fernández Cara, E. and González Burgos, Manuel and Zuazua Iriondo, Enrique
(2002)
On the Controllability of Parabolic Systems with a Nonlinear Term Involving the State and the Gradient.
SIAM Journal on Control and Optimization, 41
(3).
pp. 798-819.
ISSN 0363-0129

Abstract

We present some results concerning the controllability of a quasi-linear parabolic equation (with linear principal part) in a bounded domain of ${\mathbb R}^N$ with Dirichlet boundary conditions. We analyze the controllability problem with distributed controls (supported on a small open subset) and boundary controls (supported on a small part of the boundary). We prove that the system is null and approximately controllable at any time if the nonlinear term $f( y, \nabla y)$ grows slower than $|y| \log^{3/2}(1+ |y| + |\nabla y|) + |\nabla y| \log^{1/2}(1+ |y| + |\nabla y|)$ at infinity (generally, in this case, in the absence of control, blow-up occurs). The proofs use global Carleman estimates, parabolic regularity, and the fixed point method.